arXiv research

This paper introduces $\infty$- and $n$-fold vector bundles as special functors from the $\infty$- and $n$-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of $n$-fold vector bundles and we prove that any $n$-fold vector bundle admits a non-canonical isomorphism to a decomposed …

The paper geometrizes N-manifolds using symmetric vector bundles.

The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.

Study multiplicity-free covering of graded manifolds, proving equivalence of categories.

Vector bundles and double vector bundles, or $2$-fold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of super\-geometry and g…

Double vector bundles may be dualized in two distinct ways and these duals are themselves dual. These two dualizations generate a group, denoted $\mathscr{D}\mathscr{F}_2$, which is the symmetric group $S_3$ on three symbols. In the case of triple vector bundles the authors proved in a previous paper that the correspon…

We calculate the group of dualization operations for triple vector bundles, showing that it has order 96 and not 72 as given in Mackenzie's original treatment. The group is a nonsplit extension of S4 by the Klein group. Dualization operations are interpreted as functors on appropriate categories and are said to be equa…

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literatur…

In this paper we describe the cohomogeneity one special Lagrangian 3-folds in the cotangent bundle of the 3-sphere, also known in the physics literature as a deformed conifold. Our main result gives a global foliation of the deformed conifold by T^2-invariant special Lagrangian 3-folds, where the generic leaf is topolo…

In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n>1$, appear in various fields. The corresponding sign rule is determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The $\mathbb{Z}_2^n$-Supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinate…

Let $X$ be a complex toric Fano $n$-fold and ${\cal N}(T)$ the normalizer of a maximal torus $T$ in the group of biholomorphic authomorphisms $Aut(X)$. We call $X$ {\em symmetric} if the trivial character is a single ${\cal N}(T)$-invariant algebraic character of $T$. Using an invariant $α_G(X)$ introduced by Tian, we …

We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topol…

Cyclic covers of knots uniquely determine the original knot.

Decomposes complex manifolds with trivial canonical bundle into homogeneous structures.

The paper constructs special Lagrangian n-folds in arbitrary dimensions.

Simplicial neural networks extend graph neural networks to handle higher-order interactions.

Study of generalized vector bundles and their geometric tools.

Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.

The paper proves K-stability of special Gushel-Mukai manifolds.

Classifies equivariant vector bundles over toric manifolds.

New pushforward operation on vector pseudo-bundles creates new examples.

This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.

The study of quotient structures in multi-graded bundles, including double vector bundles.

Defines connections on parabolic vector bundles for Lie algebroids.

The paper extends positivity results from vector bundles to Kobayashi positive ones.

Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-Kähler) Hermitian metric $ω$, we introduce the concept of deformations of $X$ that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1…

The paper characterizes and contrasts knots with high 4D clasp numbers.

In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by p…

The paper extends vector bundle theory to non-Hausdorff manifolds.

Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.

In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theor…

Study on topological rigidity of ALE vector bundles with specific conditions.

Develops mixed quantization for graph vector bundles.

The paper examines the limit of harmonic flow on flat vector bundles.

Study on stable vector bundles over Gauduchon manifolds.

Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.

Defines linear weightings for vector bundles and explores their applications.

Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…

New dHYM connections found on complex vector bundles.

Theory of 2-vector bundles for smooth manifolds developed.

Griffiths' first obstruction formula for vector bundles is derived.

A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes f…

Explains differences and similarities of strictly nef and ample vector bundles.

Solves Dirac equation coupled to vector bundles.

In this paper, we prove that total space of every vector bundle with the base manifold on which the canonical isometric action acts freely, also carries a principal bundle structure. We also obtain another principal bundle based on the total space of given vector bundle.

We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main differenc…

Investigates $J$-equation on holomorphic vector bundles over Kähler manifolds.

Study biharmonic functions on vector bundles with spherical symmetry.